Smooth and piecewise linear surgery

Piecewise Linear Approximation of Smooth Compact Fibers

Achieving Linear or Quadratic Convergence on Piecewise Smooth Optimization Problems

Many problems in machine learning involve objective functions that are piecewise smooth [7] due to the occurrence of absolute values mins and maxes in their evaluation procedures. See e.g. [8]. For such function we derived in [3] first order (KKT) and second order (SSC) optimality conditions, which can be checked on the basis of a local piecewise linearization [2] that can be computed in an AD ...

Transforming the canonical piecewise-linear model into a smooth-piecewise representation

A smoothed representation (based on natural exponential and logarithmic functions) for the canonical piecewise-linear model, is presented. The result is a completely differentiable formulation that exhibits interesting properties, like preserving the parameters of the original piecewise-linear model in such a way that they can be directly inherited to the smooth model in order to determine thei...

Piecewise Constant Controlled Linear Fuzzy Systems

Conormal and Piecewise Smooth Solutions

In this paper, we show first that if a solution u of the equation i*2(l, x, u, Du, D)u = f{t, x,u, Du), where Pi{t, x,u, Du, D) is a second order strictly hyperbolic quasilinear operator, is conormal with respect to a single characteristic hypersurface S of Pi in the past and S is smooth in the past, then S is smooth and u is conormal with respect to S for all time. Second, let So arjd Si be ch...